不定方程x^(2)-90y^(2)=1与y^(2)-Dz^(2)=4的公解

On Common Solutions of Diophantine Equations x^(2)-90y^(2)=1 and y^(2)-Dz^(2)=4

主要运用同余、递归序列、Pell方程解的性质等一些初等方法,对不定方程组x^(2)-90y^(2)=1与y^(2)-Dz^(2)=4的整数解进行了研究。证明了:若p,…, p是不同的奇素数,则当D=2p…p(1≤s≤4)时,除开D为2×7×103外,不定方程组x^(2)-90y^(2)=1与y^(2)-Dz^(2)=4仅有平凡解(x,y,z)=(±19,±2,0)。

In this paper, with such elementary methods as congruence, recursive sequence and some properties of the solution to Pell equation, the integer solutions on the Diophantine equations x^(2)-90y^(2)=1 and y^(2)-Dz^(2)=4 is studied, and the result shows that: If p,…,pare diverse odd primes,D =2p…p( 1≤s ≤4), then the equations x^(2)-90y^(2) =1 and y^(2)-Dz^(2)=4 have only trivial solutions(x,y,z)=(±19,±2,0), with the exception that D =2×7×103.